32 research outputs found
Geodesic laminations revisited
The Bratteli diagram is an infinite graph which reflects the structure of
projections in a C*-algebra. We prove that every strictly ergodic unimodular
Bratteli diagram of rank 2g+m-1 gives rise to a minimal geodesic lamination
with the m-component principal region on a surface of genus g greater or equal
to 1. The proof is based on the Morse theory of the recurrent geodesics on the
hyperbolic surfaces.Comment: 13 pages, 2 figures, revised versio
Triangulations and volume form on moduli spaces of flat surfaces
In this paper, we are interested in flat metric structures with conical
singularities on surfaces which are obtained by deforming translation surface
structures. The moduli space of such flat metric structures can be viewed as
some deformation of the moduli space of translation surfaces. Using geodesic
triangulations, we define a volume form on this moduli space, and show that, in
the well-known cases, this volume form agrees with usual ones, up to a
multiplicative constant.Comment: 42 page
On embedding of the Bratteli diagram into a surface
We study C*-algebras O_{\lambda} which arise in dynamics of the interval
exchange transformations and measured foliations on compact surfaces. Using
Koebe-Morse coding of geodesic lines, we establish a bijection between Bratteli
diagrams of such algebras and measured foliations. This approach allows us to
apply K-theory of operator algebras to prove strict ergodicity criterion and
Keane's conjecture for the interval exchange transformations.Comment: final versio
Connecting geodesics and security of configurations in compact locally symmetric spaces
A pair of points in a riemannian manifold makes a secure configuration if the
totality of geodesics connecting them can be blocked by a finite set. The
manifold is secure if every configuration is secure. We investigate the
security of compact, locally symmetric spaces.Comment: 27 pages, 2 figure
Quantisations of piecewise affine maps on the torus and their quantum limits
For general quantum systems the semiclassical behaviour of eigenfunctions in
relation to the ergodic properties of the underlying classical system is quite
difficult to understand. The Wignerfunctions of eigenstates converge weakly to
invariant measures of the classical system, the so called quantum limits, and
one would like to understand which invariant measures can occur that way,
thereby classifying the semiclassical behaviour of eigenfunctions. We introduce
a class of maps on the torus for whose quantisations we can understand the set
of quantum limits in great detail. In particular we can construct examples of
ergodic maps which have singular ergodic measures as quantum limits, and
examples of non-ergodic maps where arbitrary convex combinations of absolutely
continuous ergodic measures can occur as quantum limits. The maps we quantise
are obtained by cutting and stacking
An algorithm to identify automorphisms which arise from self-induced interval exchange transformations
We give an algorithm to determine if the dynamical system generated by a
positive automorphism of the free group can also be generated by a self-induced
interval exchange transformation. The algorithm effectively yields the interval
exchange transformation in case of success.Comment: 26 pages, 8 figures. v2: the article has been reorganized to make for
a more linear read. A few paragraphs have been added for clarit